Training Thermodynamics Math The Ideal Gas Law
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The Ideal Gas Law

24 min Thermodynamics Math

The Ideal Gas Law

One of the most used equations in science — connecting pressure, volume, temperature, and amount of gas.

Ideal Gas Law

$$PV = nRT$$

$P$ = pressure (Pa), $V$ = volume (m³), $n$ = mol, $R = 8.314$ J/(mol·K), $T$ = temperature (K).

Special Cases
  • Boyle's Law (constant $T$, $n$): $P_1V_1 = P_2V_2$
  • Charles's Law (constant $P$, $n$): $V_1/T_1 = V_2/T_2$
  • Avogadro's Law (constant $P$, $T$): $V_1/n_1 = V_2/n_2$
Example 1

1 mol of gas at 300 K in a 25 L container. Find pressure.

$P = nRT/V = 1(8.314)(300)/0.025 = 99{,}768$ Pa ≈ 100 kPa ≈ 1 atm.

Example 2

A gas at 2 atm and 4 L is compressed to 1 L (constant $T$). New pressure?

$P_2 = P_1V_1/V_2 = 2(4)/1 = 8$ atm.

Example 3

A balloon has 3 L at 27°C (300 K). What volume at 127°C (400 K) at constant pressure?

$V_2 = V_1 T_2/T_1 = 3(400)/300 = 4$ L.

Practice Problems

1. Find $V$ for 2 mol at 350 K and 101325 Pa.
2. Gas at 3 atm and 6 L is expanded to 18 L. New pressure (constant T)?
3. Gas at 2 L and 200 K. Volume at 400 K (constant P)?
4. 0.5 mol gas at STP (273 K, 101325 Pa). Find volume.
5. Triple the pressure and halve the volume. What factor does T change by?
6. Can $T$ in the gas law be in °C? Why?
Show Answer Key

1. $V = 2(8.314)(350)/101325 \approx 0.0574$ m³ = 57.4 L

2. $P_2 = 3(6)/18 = 1$ atm

3. $V_2 = 2(400/200) = 4$ L

4. $V = 0.5(8.314)(273)/101325 \approx 0.0112$ m³ = 11.2 L

5. $PV = nRT$: if $P \to 3P$ and $V \to V/2$, then $T \to 3/2 \cdot T$ (factor of 1.5)

6. No — must use Kelvin. Celsius has a shifted zero, which would break the proportionality.

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